Intro Analysis for PAD Autumn 2019

*Update: The dates for 27 Nov and 4 Dec were cancelled due to UCU's industrial action.* I apologize for this.

**If you are taking this course, please register your details at <gradstud (@) maths.ox.ac.uk>. (Remove the extraneous parentheses and spaces.) This is especially important if you are being assessed.**

Course schedule: Wednesdays 10:00-12:00, starting 9 Oct and ending 4 Dec, but skipping 20 Nov (8 lectures/weeks in total).

Classroom: FRYB - G.11 (30) - Access Grid Room

Course assessment: Those taking the course for credit will be assessed based on their solutions to all of the exercises on the exercises sheet provided below as the document "PAD-exercises-20191023.pdf'. The assessment is due 9 Dec 2019 at 12:00 pm (midday/noon). Solutions are required to be texed up and the pdfs emailed to me. Assessment is only pass or fail.

Course outline: This is a short course on the basic tools in analysis covering the following topics (in order):

1. 9 Oct 2019: Function spaces (Schwartz functions, L^p spaces, Sobolev spaces) and real interpolation
2. 16 Oct 2019: The Hardy--Littlewood maximal function and Lebesgue's differentiation theorem
3. 23 Oct 2019: Birkhoff's ergodic theorem
1. 30 Oct 2019: Basics of the Fourier transform
   1. 6 Nov 2019: Effective equidistribution on the circle
   2. 13 Nov 2019: The spectral theorem
   3. No class on 20 Nov 2019. Notes on Sobolev spaces are provided below for those interested.
2. Cancelled due to industrial action: 27 Nov 2019 by Ed Crane: The Lindeberg--Levy Central limit theorem
3. Cancelled due to industrial action: 4 Dec 2019 by Ed Crane: Generalized Central limit theorems and the Berry--Essen inequality

Notes for our course will be added to the bottom of this webpage. The contents of the course may be adapted based on the audience's background.

Course background:

• For background one should be comfortable with Lebesgue integration as covered in say Stein--Shakarchi's Real Analysis or in Wheedon--Zygmund's Measure and Integral.
  • Inequalities are essential in analysis. For this topic, I recommend Steele's The Cauchy--Schwarz Masterclass. Kiran Kedlaya's notes provide a quick introduction to many of the fundamental inequalities used throughout this course.
  • One should also be comfortable with complex analysis in a single variable. Stein--Shakarchi's Complex Analysis is exemplary in this regard.

Course references: In addition to the provided notes, you may wish to consult the following references from which I will heavily draw upon for my notes.

• For lectures 1, 2, 3 and 4, the main reference are Terry Tao's notes on Fourier analysis which can be found here and here.
• For lecture 5, we will follow Everest--Ward.
• I do not know of a good reference for lecture 6. I will not provide notes at this time.
• For lectures 7 and 8, Ed will follow chapter 9 of Dudley's Real analysis and probability.
• Sobolev spaces was cancelled due to time. See the note "OdeToStein-Notes4-20190527.pdf" below if you are interested in this topic.
• Hausdorff measures and Frostman's lemma was cancelled due to time.

Please see the TCC home page for a list of other available courses.